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SAT Mathematics Review Geometry John L. Lehet
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Geometry and Measurement Geometric Notation
m
BF
The line containing point B and F
BF
The line segment with endpoints B and F
BF
The length of line segment BF
BF
The ray starting at B and extending infinitely through F
ABF
The angle formed by AB and BF
ABF
The measure of angle ABF
ABF
The triangle with vertices A, B and F
ABFG AB FG
The quadrilateral with vertices A, B, F and G AB is perpendicular to FG
Geometry and Measurement Points and Lines l A
B
unique line l, containing points A and B l A
B
M
M is the midpoint of AB, so AM = MB l C
3
D
5
E
CD = 3 and DE = 5, so CE = 3+5 = 8
Problem 1: A,B and C all lie on the same line l, if C is the midpoint of AB and AB = 12, what is AC? Problem 2: On the line l above, if CD = 4, EF = 2 and CF = 10, what is the value of DE? l C
D
E
F
Geometry and Measurement Angles in the Plane Opposite angles formed by intersecting lines are equal and are called vertical angles
Y X
Supplementary angles are Straight Angles and are equal to 180 degrees
Z W
So, X = Z and W = Y
So, X +W = 180º, X+Y = 180º, W+Z = 180º, Y+Z = 180º
Problem 1: In the above diagram, if X is equal to 40 degrees, what is the value of W? What is the value of Z? What is the value of Y? a
b
c e g
d f
h
l m
If two parallel lines (l and m) are intersected by a third line, the alternate interior angles are equal for example, e and d are alternate interior angles Problem 2: In the above diagram, l and m are parallel, name all angles that are equal to angle d? Name all angles that are supplementary to angle b?
Geometry and Measurement Triangles yº a
b xº
zº c
Equilateral Triangle equal sides (a=b=c) equal angles (x=y=z=60) Angles measure 60º
a
b xº
Problem 2: If ABC is a Right Triangle, such that m
c
yº
Isosceles Triangle two equal sides (a=b) two equal angles (x=y)
Problem 1: If ABC is an Isosceles Triangle, such that what is the m ACB?
a
ABC =
b Right Triangle one angle is 90º (a = 90) two sides are perpendicular 2 a + b2 = c2 (Pythagorean Theorem)
BAC and m
ABC is 40º,
ABC is 35º, what is the m ACB if it is not 90º?
Geometry and Measurement Triangles
30º
2x
x√3
x
45º
x√2
30º-60º-90º Triangle
5x
45º
60º
x
3x
x
4x
45º-45º-90º Triangle
Problem 1: If ABC is a Right Triangle, such that m what is the length of the longest side?
ABC is 45º and AC = 4,
3-4-5 Triangle
Geometry and Measurement Congruent Triangles triangles that have the same size and shape B
b
E b
xº A
zº
yº
yº
a
F
a
zº
c
ABC =
c
DEF
AB = DE = a BC = EF = b AC = DF = c
xº
C
D Problem 1: If ABC and DEF are congruent triangles, and AB=5 and BC=15, what is EF?
Similar Triangles triangles that have the same shape (corresponding angles are equal) E
B 8 A
xº
10
xº
zº 14
yº
4
yº D C
5 zº
7
F
ABC and DEF are similar triangles sides are proportional
Problem 2: If ABC and DEF are similar triangles, and AB=5, BC=7 and DE=15, what is EF?
Geometry and Measurement Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side B 11
xº A
AC < 8 + 11
yº
8
zº
C
Problem 1: In ABC, AB = 3 and AC = 7, can BC be 4? Can BC be 12? What are the ranges of values of BC?
Triangle Perimeter and Area Perimeter = b + a + c (sum of the three sides)
a
h
c
Area = ½bh b = base Problem 2: In the above triangle, if a=6, b= 4, c=7 and h=5, what is the perimeter? What is the area?
Geometry and Measurement Quadrilaterals b
B yº a
A
w
B
C
C
xº a
h xº
B
C l
s√2
l
s
yº
b D Parallelogram opposite angles are equal opposite sides are equal P = 2a + 2b A = bh
D w Rectangle A parallelogram with right angles P = 2w + 2l A = lw AC = BD = √(l2+w2) A
D
A
Square A rectangle with four equal sides P = 4s A = s2 AC = BD = s√2
Other Polygons A Regular Polygon is a polygon with all sides and angles equal Determine unknown angle and sides using triangles
sides interior angle sum 180º 3 360º 4 540º 5 : : 180(n-2)º n
Geometry and Measurement A
Circles
D B
O
C
O = Origin of Circle – the center OA = OB = Radius of Circle AB = Arc
AC = Diameter of Circle (twice the radius OA or OB)
the line segment AD is tangent to the circle at point A. AD touches the circle at only point A. The Diameter of a circle is twice the Radius of the circle d = 2r
The Circumference of a circle is the distance around the circle – it is analogous to perimeter of a polygon C = ∏d = 2 ∏r The Area of a circle is the amount of space within the circle – A = ∏r2 A Problem 1: Given a circle with center O and area 16∏. Points A and B are on the circle and angle OBA is 30º. Find the length of line segment AB.
O
30º
B
Geometry and Measurement Solid Figures s
h
h s s
w l Rectangular Solid Think of a Cardboard Box V = lwh SA = 2lw + 2lh + 2wh
Cube A Special Rectangular Solid in which l=w=h=s V = s3 SA = 6s2
Cylinder Think of a can of soup V = ∏r2h SA = 2∏r2 + 2∏rh
Cylinder Unfolded
Rectangular Solid Unfolded “six rectangles”
Cube Unfolded “six squares”
“circle on top circle on bottom, rectangle in the middle”
Geometry and Measurement Solid Figures h r h r
Sphere Think of a ball All radii are equal
Cone V = (1/3)∏r2h Its Volume is 1/3 of a cylinder with the same height and base
Pyramid A square at the base with four triangles V = s2h/3
Pyramid Unfolded
Problem 1: If the volume of a cube is 125 in3, what is the length of a side? What the Surface Area of the cube? Problem 2: If two cylinders have equal volume and the taller is four times higher than the shorter, what is the ratio of the radii?
“square in the middle, with four congruent triangles”
Geometry and Measurement Coordinate Geometry
Positive Slope y = 2x - 1
Negative Slope Zero Slope Undefined Slope y = -2x + 2 y=5 x=4 Two lines are parallel when their slopes are the same y = 2x + 3 is parallel to y = 2x – 7 since the slope of both lines is 2
Two lines are perpendicular when their slopes are negative reciprocals OR the product of the slopes is -1 y = -2x + 3 is perpendicular to y = (1/2)x – 7 since (-2)(1/2) = -1 OR (-2) is the negative reciprocal of (1/2) Problem 1: Give a line that is parallel to the line y = 3x – 4. Give a line that is perpendicular to it. Midpoint Formula Given two points (x1, y1) and (x2, y2) their midpoint is (xm, ym) where xm = (x1+x2)/2 and ym = (y1+y2)/2
Distance Formula Given two points (x1, y1) and (x2, y2) , their distance is d where d = √[(x1-x2)2+(y1-y2)2]
Problem 2: What is the distance of the two points (1,4) and (-1,-2)? What is their midpoint? Problem 3: If (3,2) is the midpoint of two points, one being (-1,-2), what is the other point?
Geometry and Measurement Transformations The triangle was rotated clockwise 55º
The triangle was reflected about the y-axis
The triangle was translated up and right
Translation Moves up/down and left/right
Rotation Rotates on a point Not necessarily the center
counterclockwise clockwise
Reflection Reflects along a line of symmetry
Problem 1: If a clock is rotated 90 degrees clockwise, what number will be at the top?
(-2,3)
Problem 2: If the triangle to the right is reflected about the y-axis, what are the new co-ordinates? If reflected about the x-axis, what are the new co-ordinates? Problem 3: If the triangle to the right is translated 2 units up and 3 units left, what are the new co-ordinates? If then (after translation) it is reflected about the x-axis, what are the new co-ordinates?
(-4,-1)
(3,-1)
